We consider generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be convex. Considering a lower level constraint qualification, we derive a formula for estimating the subdifferential of the value function. Finally, we establish the Fritz-John necessary optimality con...

In this article, we introduce a new class of ideal convergent sequence spaces using an infinite matrix, Musielak-Orlicz function and a new generalized difference matrix in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We also give some relations related to these sequence spaces.

Semi-supervised Support vector machine has become an increasingly popular tool for machine learning due to its wide applicability. Unlike SVM, their formulation leads to a non-smooth non-convex optimization problem. In 2005, Chapelle and Zien used a Gaussian approximation as a smooth function and presented ∇TSVM. In this paper, we propose a smooth piecewise function and research smooth piecewis...

In the theory of optimal intertemporal allocation, the assumption of a convex feasible set has played a dominant role. In recent years, several contributions have focused on the implications for this theory, when the feasible set does not have the convexity property. (See, in particular, Skiba (1978), Majumdar and Mitra (1982, 1983), Dechert and Nishimura (1983), Majumdar and Nermuth (1982), an...

The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, varian...

Neural networks with ReLU activation function have been shown to be universal approximators and learn mapping as non-smooth functions. Recently, there is considerable interest in the use of neural applications such optimal control. It well-known that optimization involving non-convex, functions are computationally intensive limited convergence guarantees. Moreover, choice hyper-parameters used ...

In this paper, we study ε-generalized weak s...

Khrapchenko’s classical lower bound n on the formula size of the parity function f can be interpreted as designing a suitable measure of subrectangles of the combinatorial rectangle f−1(0)× f−1(1). Trying to generalize this approach we arrived at the concept of convex measures. We prove the negative result that convex measures are bounded by O(n) and show that several measures considered for pr...

The problem of minimizing f̃ = f +p over some convex subset of a Euclidean space is investigated, where f(x) = x Ax + b x is a strictly convex quadratic function and |p| is only assumed to be bounded by some positive number s. It is shown that the function f̃ is strictly outer γ-convex for any γ > γ∗, where γ∗ is determined by s and the smallest eigenvalue of A. As consequence, a γ∗-local minimal...

In recent years, a rapidly increasing number of applications in practice requires optimizing non-convex objectives, like training neural networks, learning graphical models, maximum likelihood estimation. Though simple heuristics such as gradient descent with very few modifications tend to work well, theoretical understanding is very weak. We consider possibly the most natural class of non-conv...